A sieve $S$ on $U$ in the category $\mathcal{O}(X)$ is principal iff the corresponding subfunctor $S\subset 1_U\cong{\rm Hom}(-,U)$ is a sheaf.

Question

This is Exercise II.1 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, this is new to MSE.

The Details:

On p. 36, ibid. . . .

Definition 0: For an arbitrary small category $\mathbf{C}$, a subfunctor of $P:\mathbf{C}^{{\rm op}}\to\mathbf{Sets}$ is defined to be another functor $Q:\mathbf{C}^{{\rm op}}\to\mathbf{Sets}$ with each $QC$ a subset of $PC$ and $Qf: QD\to QC$ a restriction of $Pf$, for all $C\stackrel{f}{\to} D\in {\rm Mor}(\mathbf{C})$.

On p. 37, ibid. . . .

Definition 1: Given an object $C$ in the category $\mathbf{C}$, a sieve on $C$ [. . .] is a set $S$ of arrows with codomain $C$ such that

$f \in S$ and the composite $fh$ is defined implies $fh \in S$.

Let $X$ be a topological space.

On p. 66, ibid. . . .

Definition 2: A sheaf of sets $F$ on a topological space $X$ is a functor $F:\mathcal{O}(X)^{{\rm op}}\to\mathbf{Sets}$ such that each open covering $U=\bigcup_iU_i, i\in I$, of open subsets of $U$ of $X$ yields an equaliser diagram

$$ FU\stackrel{e}{\dashrightarrow}\prod_{i\in I}FU_i\overset{p}{\underset{q}{\rightrightarrows}}\prod_{i,j\in I}(U_i\cap U_j),$$

where for $t\in FU,$ $e(t)=\{ t\rvert_{U_i}\mid i\in I\}$ and for a family $t_i\in FU_i$,

$$p\{ t_i\}=\{t_i\rvert_{(U_i\cap U_j)}\}\quad\text{ and }\quad q\{ t_i\}=\{t_j\rvert_{(U_i\cap U_j)}\}.$$

On p. 70, ibid. . . .

Definition 3: Recall from $\S I.4$ that a sieve $S$ on $U$ in this category is defined to be a subfunctor of ${\rm Hom}( - , U).$ Replacing the sieve $S$ by the set (call it $S$ again) of all those $V \subset U$ with $SV = 1$, we may also describe a sieve on $U$ as a subset $S\subset \mathcal{O}(U)$ of objects such that $V_0\subset V\in S$ implies $V_0 \in S$. Each indexed family $\{V_i \subset U\mid i \in I\}$ of subsets of $U$ generates (= "spans") a sieve $S$ on $U$; namely, the set $S$ consisting of all those open $V$ with $V \subset V_i$ for some $i$; in particular, each $V_0 \subset U$ determines a principal sieve ($V_0$) on $U$, consisting of all $V$ with $V \subset V_0$.

Here $\mathcal{O}(U)$ is the set of open subsets of $U$.

It is then claimed that . . .

It is not difficult to see that a sieve $S$ on $U$ is principal iff the subfunctor $S$ of $\mathbf{y}(U)$ is a subsheaf. (Exercise II.1.)

Here

$$\begin{align} \mathbf{y}: \mathbf{C} &\to \mathbf{Sets}^{\mathbf{C}^{{\rm op}}}, \\ C &\mapsto {\rm Hom}_{\mathbf{C}}( - , C) \end{align}$$

is the Yoneda embedding.

The Question:

Exercise II.1: Show that a sieve $S$ on $U$ in the category $\mathcal{O}(X)$ is principal iff the corresponding subfunctor $S\subset 1_U \cong {\rm Hom}( - ,U)$ is a sheaf.

Thoughts:

$(\Rightarrow)$ Suppose a sieve $S$ on $U$ in the category $\mathcal{O}(X)$ is principal. Then, if I understand this correctly, $S=(V_0)$ consists of all $V$ such that $V\stackrel{?}{\subseteq}V_0$.

Then what? I'm not sure I understand Definition 2.

$(\Leftarrow)$ Again, I'm not sure of Definition 2, so I can't really make a start on this.

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Please help :)

Answer 1

Let $S$ be a sieve on $U$ in $\newcommand\calO{\mathcal{O}}\calO(X)$.

We want to show

$S$ is principal if and only if $S$ is a sheaf on $\calO(X)$.

Principal implies sheaf

First, suppose $S$ is principal, i.e., generated by $V_0\subseteq U$ for some $V_0$.

Let $W_i$, $i\in I$ be a cover of $W$. We need to show that $$ SW \to \prod_i SW_i \rightrightarrows \prod_{i,j} S(W_i\cap W_j) $$ is an equalizer diagram. Now for any set $V$, $SV$ is either empty (if $V\not\subseteq V_0$) or $SV$ contains the morphism $V\subseteq U$ if $V\subseteq V_0$.

Then if for some $i$, one of the $SW_i$ is empty, the product in the middle is empty, and $SW$ is empty, since there is $x\in W_i\setminus V\subseteq W\setminus V$, and the diagram becomes $$\varnothing\to\varnothing \rightrightarrows \varnothing,$$ which is immediately an equalizer.

On the other hand, if $SW_i$ is nonempty for all $i$, then $W_i\subseteq V_0$ for all $i$, and thus, since $W=\bigcup_i W_i$, $W\subseteq V_0$. Thus the diagram becomes $$\{*\}\to \{*\} \rightrightarrows \{*\},$$ which is again immediately an equalizer.

Thus principal sieves are sheaves.

Sheaf implies principal

Now suppose $S$ is a sheaf on $\calO(X)$. Consider the collection $$\mathcal{W} = \{W : S(W) \ne\varnothing \}$$ Clearly $\mathcal{W}$ covers $V:=\bigcup \mathcal{W}$. Then since $$ SV \to \prod_{W\in\mathcal{W}} SW \rightrightarrows \prod_{W,W'\in\mathcal{W}} S(W\cap W') $$ is an equalizer, and since $S(W)$, $S(W\cap W')$ are all nonempty, and thus one element sets, we have that $$ SV\to \{*\} \rightrightarrows \{*\} $$ is an equalizer, so $SV$ is a one element set containing $V\subseteq U$.

Then by construction, $SW\ne\varnothing$ if and only if $W\subseteq V$, so $S$ is the principal sieve generated by $V$. $\blacksquare$